tmua.me Medium Set D Paper 2 · Question 20

For a real constant $a$ , define $f(x)=x^3+ax$ for all real $x$ . It is known that

f(x+1)>f(x)

for every real $x$ .

Consider the statements

I. $a>-\dfrac14$ .
II. For all real $u<v$ , we have $f(u)<f(v)$ .
III. Whenever $u$ and $v$ are distinct real numbers with $f(u)=f(v)$ , we have $|u-v|<1$ .

Which statements must be true?

\mathrm{I}

only

\mathrm{II}

only

\mathrm{III}

only

\mathrm{I}

and

\mathrm{II}

only

\mathrm{I}

and

\mathrm{III}

only

\mathrm{II}

and

\mathrm{III}

only

\mathrm{I}

\mathrm{II}

and

\mathrm{III}

examday